Physics 221A Fall 2019 Notes 21 Time Reversal

[Pages:19]Copyright c 2019 by Robert G. Littlejohn

Physics 221A Fall 2019 Notes 21

Time Reversal

1. Introduction

We have now considered the space-time symmetries of translations, proper rotations, and spatial inversions (that is, improper rotations) and the operators that implement these symmetries on a quantum mechanical system. We now turn to the last of the space-time symmetries, namely, time reversal. As we shall see, time reversal is different from all the others, in that it is implemented by means of antiunitary transformations.

2. Time Reversal in Classical Mechanics

Consider the classical motion of a single particle in three-dimensional space. Its trajectory x(t) is a solution of the equations of motion, F = ma. We define the time-reversed classical motion as x(-t). It is the motion we would see if we took a movie of the original motion and ran it backwards. Is the time-reversed motion also physically allowed (that is, does it also satisfy the classical equations of motion)?

The answer depends on the nature of the forces. Consider, for example, the motion of a charged particle of charge q in a static electric field E = -, for which the equations of motion are

m

d2x dt2

=

qE(x).

(1)

If x(t) is a solution of these equations, then so is x(-t), as follows from the fact that the equations

are second order in time, so that the two changes of sign coming from t -t cancel. However, this

property does not hold for magnetic forces, for which the equations of motion include first order

time derivatives:

m

d2x dt2

=

q c

dx dt

?B(x).

(2)

In this equation, the left-hand side is invariant under t -t, while the right-hand side changes

sign. For example, in a constant magnetic field, the sense of the circular motion of a charged

particle (clockwise or counterclockwise) is determined by the charge of the particle, not the initial

conditions, and the time-reversed motion x(-t) has the wrong sense. We see that motion in a given

electric field is time-reversal invariant, while in a magnetic field it is not.

Links to the other sets of notes can be found at: .

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Notes 21: Time Reversal

We must add, however, that whether a system is time-reversal invariant depends on the defini-

tion of "the system." In the examples above, we were thinking of the system as consisting of a single

charged particle, moving in given fields. But if we enlarge "the system" to include the charges that

produce the fields (electric and magnetic), then we find that time-reversal invariance is restored,

even in the presence of magnetic fields. This is because when we set t -t, the velocities of all

the particles change sign, so the current does also. But this change does nothing to the charges of

the particles, so the charge density is left invariant. Thus, the rules for transforming charges and

currents under time reversal are

, J -J.

(3)

But according to Maxwell's equations, this implies the transformation laws

E E, B -B,

(4)

for the electromagnetic field under time reversal. With these rules, we see that time-reversal invariance is restored to Eq. (2), since there are now two changes of sign on the right hand side.

Thus we have worked out the basic transformation properties of the electromagnetic field under time reversal, and we find that electromagnetic effects are overall time-reversal invariant. We have shown this only in classical mechanics, but it is also true in quantum mechanics.

Similarly, in quantum physics we are often interested in the time-reversal invariance of a given system, such as an atom interacting with external fields. The usual point of view is to take the external fields as just given, and not to count them as part of the system. Under these circumstances the atomic system is time-reversal invariant if there are no external magnetic fields, but time-reversal invariance is broken in their presence. On the other hand, the atom generates its own, internal, magnetic fields, such as the dipole fields associated with the magnetic moments of electrons or nuclei, or the magnetic field produced by the moving charges. Since these fields are produced by charges that are a part of "the system," however, they do not break time-reversal invariance. We summarize these facts by saying that electromagnetic effects are time-reversal invariant in isolated systems.

It turns out the same is true for the strong forces, a fact that is established experimentally. The weak forces do, however, violate time-reversal invariance (or at least CP-invariance) at a small level. We shall say more about such violations later in these notes.

3. Time Reversal and the Schro?dinger Equation

Let us consider the quantum analog of Eq. (1), that is, the motion of a charged particle in a

given electric field. The Schr?odinger equation in this case is

i?h

(x, t

t)

=

-

?h2 2m

2

+

q(x)

(x, t).

(5)

Suppose (x, t) is a solution of this equation. Following what we did in the classical case, we ask

if (x, -t) is also a solution. The answer is no, for unlike the classical equations of motion (1), the

Notes 21: Time Reversal

3

Schr?odinger equation is first order in time, so the left hand side changes sign under t -t, while the right hand side does not. If, however, we take the complex conjugate of Eq. (5), then we see that (x, -t) is a solution of the Schr?odinger equation, since the complex conjugation changes the sign of i on the left hand side, which compensates for the change in sign from t -t.

Altogether, we see that if we define the time-reversed motion in quantum mechanics by the rule

r(x, t) = (x, -t),

(6)

where the r-subscript means "reversed," then charged particle motion in a static electric field is time-

reversal invariant. We can see already from this example that time reversal in quantum mechanics

is represented by an antilinear operator, since a linear operator is unable to map a wave function

into its complex conjugate.

Similarly, the quantum analog of Eq. (2) is the Schr?odinger equation for a particle in a magnetic

field,

i?h

(x, t

t)

=

1 2m

- i?h -

q c

A(x)

2

(x, t).

(7)

In this case if (x, t) is a solution, it does not follow that (x, -t) is a solution, because of the

terms that are linear in A. These terms are purely imaginary, and change sign when we complex conjugate the Schr?odinger equation. But (x, -t) is a solution in the reversed magnetic field, that

is, after the replacement A -A. This is just as in the classical case.

4. The Time-Reversal Operator

The definition (6) of the time-reversed wave function applies to a spinless particle moving in

three-dimensions. We shall be interested in generalizing it to other systems, such as multiparticle

systems with spin, as well as preparing the ground for generalizations to relativistic systems and

quantum fields. If |(t) is a time-dependent state vector of a system that satisfies the Schr?odinger

equation

i?h

t

|(t)

= H|(t) ,

(8)

then on analogy with Eq. (6) we shall write the time-reversed state as

|r(t) = |(-t) ,

(9)

where , the time-reversal operator, is to be defined for different systems based on certain postulates that we shall require of it. The operator by itself does not involve time, as we see from Eq. (6), where it has the effect of complex conjugating the wave function; rather, it is a mapping that takes kets into other kets. In particular, setting t = 0 in Eq. (9),

|r(0) = |(0) ,

(10)

we see that maps the initial conditions of the original motion into the initial conditions of the time-reversed motion.

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Notes 21: Time Reversal

We obtain a set of postulates for as follows. First, since probabilities should be conserved

under time reversal, we require

= 1.

(11)

Next, in classical mechanics, the initial conditions of a motion x(t) transform under time reversal according to (x0, p0) (x0, -p0), so we postulate that the time-reversal operator in quantum mechanics should satisfy the conjugation relations,

x = x, p = -p.

(12)

This should hold in systems where the operators x and p are meaningful. In such systems, these

requirements imply

L = -L,

(13)

where L = x?p is the orbital angular momentum. As for spin angular momentum, we shall postulate that it transform in the same way as orbital angular momentum,

S = -S.

(14)

This is plausible in view of a simple classical model of a spin, in which a particle like an electron

is seen as a small charged sphere spinning on its axis. The rotation produces both an angular

momentum and a magnetic moment. This model has flaws and cannot be taken very seriously, but

at least it does indicate that if we reverse the motion of the charges on the sphere, both the angular

momentum and the magnetic moment should reverse. Accepting both Eqs. (13) and (14), we see

that we should have

J = -J,

(15)

for all types of angular momentum.

5. Cannot Be Unitary

It turns out that the conjugation relations (12) cannot be satisfied by any unitary operator. For if we take the canonical commutation relations,

[xi, pj] = ih? ij,

(16)

and conjugate with , we find

[xi, pj] = -[xi, pj] = -i?h ij = (i?h ij).

(17)

The quantity i?h ij is just a number, so if is unitary it can be brought through to cancel , and we obtain a contradiction. Thus, we are forced to conclude that the time-reversal operator must be antilinear, so that the imaginary unit i on the right-hand side of Eq. (17) will change into -i when is pulled through it.

Notes 21: Time Reversal

5

6. Wigner's Theorem

A famous theorem proved by Wigner says that if we have a mapping of a ket space onto itself,

taking, say, kets | and | into kets | and | , such that the absolute values of all scalar products

are preserved, that is, such that

| | | = | | |

(18)

for all | and | , then, to within inessential phase factors, the mapping must be either a linear unitary operator or an antilinear unitary operator. The reason Wigner does not demand that the scalar products themselves be preserved (only their absolute values) is that the only quantities that are physically measurable are absolute squares of scalar products. These are the probabilities that are experimentally measurable. This theorem is discussed in more detail by Messiah, Quantum Mechanics, in which a proof is given. (See also Steven Weinberg, The Quantum Theory of Fields I.) Its relevance for the discussion of symmetries in quantum mechanics is that a symmetry operation must preserve the probabilities of all experimental outcomes, and thus all symmetries must be implemented either by unitary or antiunitary operators. In fact, all symmetries except time reversal (translations, proper rotations, parity, and others as well) are implemented by unitary operators. Time reversal, however, requires antiunitary operators.

7. Properties of Antilinear Operators

Since we have not encountered antilinear operators before, we now make a digression to discuss their mathematical properties. We let E be the ket space of some quantum mechanical system. In the following general discussion we denote linear operators by L, L1, etc., and antilinear operators by A, A1, etc. Both linear and antilinear operators are mappings of the ket space onto itself,

L : E E,

A : E E,

(19)

but they have different distributive properties when acting on linear combinations of kets:

L c1|1 + c2|2 = c1 L|1 + c2 L|2 A c1|1 + c2|2 = c1 A|1 + c2 A|2

(20a) (20b)

(see Eqs. (1.34)). In particular, an antilinear operator does not commute with a constant, when the latter is regarded as a multiplicative operator in its own right. Rather, we have

Ac = cA.

(21)

It follows from these definitions that the product of two antilinear operators is linear, and the product of a linear with an antilinear operator is antilinear. More generally, a product of operators is either linear or antilinear, depending on whether the number of antilinear factors is even or odd, respectively.

6

Notes 21: Time Reversal

We now have to rethink the entire Dirac bra-ket formalism, to incorporate antilinear operators. To begin, we define the action of antilinear operators on bras. We recall that a bra, by definition, is a complex-valued, linear operator on kets, that is, a mapping,

bra : E C,

(22)

and that the value of a bra acting on a ket is just the usual scalar product. Thus, if | is a bra,

then we have

| | = | .

(23)

We now suppose that an antilinear operator A is given, that is, its action on kets is known, and we wish to define its action on bras. For example, if | is a bra, we wish to define |A. In the case of linear operators, the definition was

|L | = | L| .

(24)

Since the positioning of the parentheses is irrelevant, it is customary to drop them, and to write simply |L| . In other words, we can think of L as acting either to the right or to the left. However, the analogous definition for antilinear operators does not work, for if we try to write

|A | = | A| ,

(25)

then |A is indeed a complex-valued operator acting on kets, but it is an antilinear operator, not a linear one. Bras are supposed to be linear operators. Therefore we introduce a complex conjugation to make |A a linear operator on kets, that is, we set

|A | = | A| . (26)

This rule is easiest to remember in words: we say that in the case of an antilinear operator, it does matter whether the operator acts to the right or to the left in a matrix element, and if we change the direction in which the operator acts, we must complex conjugate the matrix element. In the case of antilinear operators, parentheses are necessary to indicate which direction the operator acts. The parentheses are awkward, and the fact is that Dirac's bra-ket notation is not as convenient for antilinear operators as it is for linear ones.

Next we consider the definition of the Hermitian conjugate. We recall that in the case of linear operators, the Hermitian conjugate is defined by

L| = |L ,

(27)

for all kets | , or equivalently by

|L| = |L| ,

(28)

for all kets | and | . Here the linear operator L is assumed given, and we are defining the new linear operator L. The definition (27) also works for antilinear operators, that is, we set

A| = |A .

(29)

Notes 21: Time Reversal

7

We note that by this definition, A is an antilinear operator if A is antilinear. Now, however, when we try to write the analog of (28), we must be careful about the parentheses. Thus, we have

| A| = |A | .

(30)

This rule is also easiest to remember in words. It is really a reconsideration of the rule stated in Sec. 1.13, that the Hermitian conjugate of any product of complex numbers, kets, bras and operators is obtained by reversing the order and taking the Hermitian conjugate of all factors. This rule remains true when antilinear operators are in the mix, but the parentheses indicating the direction in which the antilinear operator acts must also be reversed at the same time that A is changed into A or vice versa. That is, the direction in which the antilinear operator acts is reversed.

The boxed equations (21) and (26) summarize the principal rules for antilinear operators that differ from those of linear operators.

8. Antiunitary Operators

We wrote down Eq. (11) thinking that it would require probabilities to be preserved under . This would certainly be true if were unitary, but since we now know must be antilinear, we should think about probability conservation under antilinear transformations.

We define an antiunitary operator A as an antilinear operator that satisfies

AA = AA = 1.

(31)

We note that the product AA or AA is a linear operator, so this definition is meaningful. Just like unitary operators, antiunitary operators preserve the absolute values of scalar products, as indicated by Wigner's theorem. To see this, we let | and | be arbitrary kets, and we set | = A| , | = A| , where A is antiunitary. Then we have

| = |A A| = | AA| = | ,

(32)

where we reverse the direction of A in the second equality and use AA = 1 in the third. Antiunitary operators take scalar products into their complex conjugates, and Eq. (18) is satisfied. Thus, we were correct in writing down Eq. (11) for probability conservation under time reversal.

9. The LK Decomposition

Given an antilinear operator A of interest, it is often convenient to factor it into the form

A = LK,

(33)

where L is a linear operator and K is a particular antilinear operator chosen for its simplicity. The idea is that K takes care of the antilinearity of A, while L takes care of the rest. The choices made for K are usually of the following type.

8

Notes 21: Time Reversal

Let Q stand for a complete set of commuting observables (a single symbol Q for all operators

in the set). Let n be the collective set of quantum numbers corresponding to Q, so that the basis

kets in this representation are |n . The index n can include continuous quantum numbers as well

as discrete ones. Then we define a particular antilinear operator KQ by requiring, first, that KQ be antilinear, and second, that

KQ|n = |n .

(34)

Notice that the definition of KQ depends not only on the operators Q that make up the representation, but also the phase conventions for the eigenkets |n . If KQ were a linear operator, Eq. (34) would imply KQ = 1; but since KQ is antilinear, the equation KQ = 1 is not only not true, it is meaningless, since it equates an antilinear operator to a linear one. But Eq. (34) does completely specify KQ, for if | is an arbitrary ket, expanded according to

| = cn|n ,

(35)

n

then

KQ| = cn|n ,

(36)

n

where we use Eqs. (20b) and (34). Thus, the action of KQ on an arbitrary ket is known. The effect

of KQ is to bring about a complex conjugation of the expansion coefficients in the Q representation.

These expansion coefficients are the same as the wave function in the Q representation; thus, in

wave function language in the Q representation, KQ just maps the wave function into its complex conjugate.

Consider, for example, the ket space for a spinless particle in three dimensions. Here we can

work in the position representation, in which Q = x and in which the basis kets are |x . Then we

define the antilinear operator Kx by

Kx|x = |x ,

(37)

so that if | is an arbitrary ket and (x) its wave function, then

Kx| = Kx d3x |x x| = Kx d3x |x (x) = d3x |x (x).

(38)

Thus, (x) is mapped into (x).

The operator KQ looks simple in the Q-representation. It may of course be expressed in other representations, but then it no longer looks so simple. For example, Kx is not as simple in the momentum representation as in the configuration representation (an explicit expression for Kx in the momentum representation will be left as an exercise).

It follows from the definition (34) that KQ satisfies

KQ2 = 1.

(39)

(Just multiply Eq. (34) by KQ and note that KQ2 is a linear operator, so that Eq. (39) is meaningful.)

The operator KQ also satisfies

KQ = KQ ,

(40)

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